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Properties

Label 2-442-221.16-c1-0-13
Degree $2$
Conductor $442$
Sign $0.748 - 0.663i$
Analytic cond. $3.52938$
Root an. cond. $1.87866$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 0.866i)2-s + (0.340 − 0.196i)3-s + (−0.499 + 0.866i)4-s − 1.77i·5-s + (0.340 + 0.196i)6-s + (2.18 + 1.26i)7-s − 0.999·8-s + (−1.42 + 2.46i)9-s + (1.53 − 0.888i)10-s + (2.98 − 1.72i)11-s + 0.393i·12-s + (2.99 − 2.01i)13-s + 2.52i·14-s + (−0.349 − 0.604i)15-s + (−0.5 − 0.866i)16-s + (4.08 + 0.551i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.196 − 0.113i)3-s + (−0.249 + 0.433i)4-s − 0.794i·5-s + (0.138 + 0.0802i)6-s + (0.824 + 0.476i)7-s − 0.353·8-s + (−0.474 + 0.821i)9-s + (0.486 − 0.280i)10-s + (0.899 − 0.519i)11-s + 0.113i·12-s + (0.830 − 0.557i)13-s + 0.673i·14-s + (−0.0901 − 0.156i)15-s + (−0.125 − 0.216i)16-s + (0.991 + 0.133i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 442 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.748 - 0.663i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 442 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.748 - 0.663i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(442\)    =    \(2 \cdot 13 \cdot 17\)
Sign: $0.748 - 0.663i$
Analytic conductor: \(3.52938\)
Root analytic conductor: \(1.87866\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{442} (237, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 442,\ (\ :1/2),\ 0.748 - 0.663i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.79197 + 0.680357i\)
\(L(\frac12)\) \(\approx\) \(1.79197 + 0.680357i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (-2.99 + 2.01i)T \)
17 \( 1 + (-4.08 - 0.551i)T \)
good3 \( 1 + (-0.340 + 0.196i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + 1.77iT - 5T^{2} \)
7 \( 1 + (-2.18 - 1.26i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.98 + 1.72i)T + (5.5 - 9.52i)T^{2} \)
19 \( 1 + (3.66 - 6.34i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.147 + 0.0854i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.57 + 2.06i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 9.12iT - 31T^{2} \)
37 \( 1 + (9.10 - 5.25i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.111 + 0.0644i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.65 - 4.60i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 2.05T + 47T^{2} \)
53 \( 1 + 4.26T + 53T^{2} \)
59 \( 1 + (-0.0723 + 0.125i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (13.4 + 7.76i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.08 - 7.07i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (4.14 + 2.39i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 9.40iT - 73T^{2} \)
79 \( 1 - 2.76iT - 79T^{2} \)
83 \( 1 - 6.67T + 83T^{2} \)
89 \( 1 + (7.64 + 13.2i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.50 + 3.17i)T + (48.5 + 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.43096191962163195634108880430, −10.34678323896907466250882086620, −9.013410005463232474937120423575, −8.136787239863380331361326403610, −8.056590718324234691075064696791, −6.26980596933178375449487035937, −5.57017860929858916966997861734, −4.61564750928493783209618682902, −3.38230867010125758032690870366, −1.58808483659907408427536002687, 1.41760040079499472131143723841, 3.00444931019149873601740230754, 3.92965288864730175314098993390, 4.99179078441099472500891390661, 6.45607942835437656949293215018, 7.05344880576985201451818115336, 8.600285309802190125740947008326, 9.195042868796925179992049378272, 10.44026725338370034057808732940, 10.98598226643357797152079967958

Graph of the $Z$-function along the critical line